I would agree with Cees - craigslist would be best private option.
However, if you just want a full service
company to handle it all, pick-up, store and drop off - I know this company very well. I have worked with them for years - usually long term, but they also do short durations.
http://www.thecorporategarage.netYou can drop off to Perris, CA (about 90-100 miles from LA) or they pick up in LA area often. You can talk to Debbie or Mike Boggs.
It's also possible to drop off to Costa Mesa and Mike will trailer it to Perris.